### Geometric structures on complex manifolds: talks and abstracts

#### 3-7 October 2011, Laboratory of Algebraic Geometry, Steklov Institute, Moscow

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#### Dmitri Alekseevsky (Masaryk University in Brno) Lorentzian manifolds with large isometry group

I give a survey of results about isometry group of Lorentzian manifolds and will describe some classes of homogeneous Lorentzian manifolds including homogeneous manifolds of a semisimple Lie group and manifolds with weakly irreducible isotropy group.

#### Semyon Alesker (Tel Aviv University) Quaternionic Monge-Ampere equations and HKT-geometry.

A notion of quaternionic Monge-Ampere equation will be introduced. These are non-linear second order elliptic equations which make sense on so called hypercomplex manifolds, in particular on the flat quaternionic space. They admit an interpretation in the framework of Hyper Kahler with Torsion (HKT) geometry (to be explained in the talk). We formulate a quaternionic version of the Calabi conjecture, and state a number of partial results towards its proof. Part of the results are joint with M. Verbitsky.

#### Jürgen Berndt (King's College London) Polar actions on symmetric spaces

An isometric action of a connected Lie group $H$ on a Riemannian manifold $M$ is called polar if there exists a connected closed submanifold $\Sigma$ of $M$ such that $\Sigma$ meets each orbit of the action and intersects it orthogonally. An elementary example of a polar action comes from the standard representation of $SO_n$ on ${\mathbb R}^n$. Further examples of polar actions can be constructed from Riemannian symmetric spaces. Let $M = G/K$ be a Riemannian symmetric space and denote by $o$ a fixed point of the $K$-action on $M$. Then the isotropy representation $\pi : K \to O(T_oM)$ of $K$ on the tangent space $T_oM$ of $M$ at $o$ induces a polar action. Dadok established in 1985 a remarkable, and mysterious, relation between polar actions on Euclidean spaces and Riemannian symmetric spaces. He proved that for every polar action on ${\mathbb R}^n$ there exists a Riemannian symmetric space $M = G/K$ with $\dim M = n$ such that the orbits of the action on ${\mathbb R}^n$ and the orbits of the $K$-action on $T_oM$ are the same via a suitable isomorphism ${\mathbb R}^n \to T_oM$.

Soon afterwards an attempt was made to classify polar actions on symmetric spaces. For irreducible symmetric spaces of compact type the final step for a complete classification appears to have been just completed by Kollross using yet unpublished work of Lytchak on polar foliations. In the talk I want to focus on symmetric spaces of noncompact type. For actions of reductive groups one can use the concept of duality between symmetric spaces of compact type and of noncompact type. However, new examples and phenomena arise from the geometry induced by actions of parabolic subgroups, for which there is no analogon in the compact case. I plan to discuss the main difficulties one encounters here and some partial solutions. The only complete classification known so far has just been obtained in joint work with Jos\'{e} Carlos D\'{i}az-Ramos for the complex hyperbolic plane.

#### Roger Bielawski (University of Leeds) Hypercomplex and pluricomplex geometry

I'll will describe a new type of geometric structure on complex manifolds. It can be viewed as a deformation of hypercomplex structure, but it also leads to a special type of hypercomplex geometry. These structures have both algebro-geometric and differential-geometric descriptions, and there are interesting examples arising from physics.

#### Gil Cavalcanti (Utrecht University) Generalized Kahler structures on moduli space of instantons

We show how the reduction procedure for generalized Kahler structures can be used to recover Hitchin's results about the existence of a generalized Kahler structure on the moduli space of instantons on bundle over a generalized Kahler manifold. In this setup the proof follows closely the proof of the same claim for the Kahler case and clarifies some of the stranger considerations from Hitchin's proof.

#### Vicente Cortés (Hamburg University) From cubic polynomials to complete quaternionic Kahler manifolds

I will explain two supergravity constructions, which allow to construct certain special Riemannian manifolds starting with other special Riemannian manifolds. We show that the resulting manifolds are complete if the original manifolds are. By composition of the two constructions we obtain complete quaternionic Kahler manifolds out of certain cubic hypersurfaces. At the end I will formulate two open problems concerning such hypersurfaces.

The talk is based on arXiv:1101.5103 (hepth, mathdg).

#### Isabel Dotti (University of Cordoba, Argentina) Some restrictions on existence of abelian complex structures

We describe the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. More precisely, we consider a distinguished class of Lie algebras admitting abelian complex structures given by abelian double products. The structure of these Lie algebras can be described in terms of a pair of commutative associative algebras satisfying a compatibility condition. We will show that when g is a Lie algebra with an abelian complex structure J, and g decomposes as g = u + Ju, with u an abelian subalgebra, then g is an abelian double product.

Joint work with A. Andrada and M. L. Barberis

#### Dmitry Egorov (Yakutsk) A new equation on the low-dimensional Calabi-Yau metrics.

In this talk I wil introduce a new equation on the compact Kahler manifolds. Solution of this equation corresponds to the Calabi--Yau metric. New equation describes deformation of complex structure, while Monge--Ampere equation describes deformation of symplectic structure.

#### Anna Fino (Torino University) Special Hermitian structures and symplectic geometry

Symplectic forms taming complex structures on compact manifolds are strictly related to a special type of Hermitian metrics, known in the literature as "strong Kaehler with torsion" metrics. I will present general results on "strong Kahler with torsion" metrics, their link with symplectic geometry and more in general with generalized complex gometry. Moreover, I will show for certain 4-dimensional non-Kaehler symplectic 4-manifolds some recent results about the Calabi-Yau equation in the context of symplectic geometry.

#### Akira Fujiki (Osaka) Joyce twistor space and the associated K\"ahler class

Fix a smooth action of a real two-torus on the connected sum of m copies of complex projective plane. Then the invariant self-dual structures constructed by Joyce, and hence the associated twistor spaces, depend on real (m-1)-dimensional parameter. We can then associate each twistor space a Kaehler class on a fixed open rational surface, which makes the moduli space of Joyce twistor spaces a domain of the projectified K\"ahler cone of the surface. This result then gives a nice description of the families of anti-self-dual bihermitian structures on hyperbolic Inoue surfaces constructed previously with Pontecorvo.

#### Paul Gauduchon (Ecole Polytechnique) Almost complex structures on quaternion-Kaehler manifolds

The aim of this lecture is to show that, apart from the complex Grassmannians $Gr_2(C^{n+2})$, the compact quaternionic quaternion-Kaehler manifolds of positive type admit no almost complex structure, even in the weak sense (joint work with Andrei Moroianu and Uwe Semmelmann).

#### Ryushi Goto (Osaka) Deformations of L.C.K. structures and generalized Kaehler structures

A notion of geometric structures which includes locally conformally Kaehler and generalized Kaehler structures will be introduced. Unobstructed deformations of the structures are discussed from the unified view point. Deformations on non-Kaehler manifolds such as Vaisman manifolds and Inoue surfaces will be given.

#### Gueo Grantcharov (Florida Intl. University) Calibrations in hyperkaehler geometry

We describe a family of calibrations arising naturally on a hyperkaehler manifold M. These calibrationscalibrate the holomorphic Lagrangian, holomorphic isotropic and holomorphic coisotropic subvarieties. When M is an HKT (hyperkahler with torsion) manifold with holonomy SL(n,H), we construct another family of calibrations, which calibrate holomorphic Lagrangian and holomorphic coisotropic subvarieties. They are (generally speaking) not parallel with respect to any torsionless connection on M. We note also that there are examples of complex isotropic submanifolds in SL(n, H) manifolds with HKT structure, which can not be calibrated by any form, unlike the Kaehler case.

#### Keizo Hasegawa (Niigata University) Locally conformally Kaehler structures on homogeneous spaces

A homogeneous Hermitian manifold M with its homogeneous Hermitian structure h, defining a locally conformally Kaehler structure w is called a homogeneous locally conformally Kaehler or shortly a homogeneous l.c.K. manifold. If a simply connected homogeneous l.c.K. manifold M=G/H, where G is a connected Lie group and H a closed subgroup of G, admits a free action of a discrete subgroup D of G from the left, then a double coset space D\G/H is called a locally homogeneous l.c.K. manifold. We discuss explicitly homogeneous and locally homogeneous l.c.K. structures on Hopf surfaces and Inoue surfaces, and their deformations. We also classify all complex surfaces admitting locally homogeneous l.c.K. structures.

We show as a main result a structure theorem of compact homogeneous l.c.K. manifolds, asserting that it has a structure of a holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus. As an application of the theorem, we see that only compact homogeneous l.c.K. manifolds of complex dimension 2 are Hopf surfaces of homogeneous type. We also see that there exist no compact complex homogeneous l.c.K. manifolds; in particular neither complex Lie groups nor complex paralellizable manifolds admit their compatible l.c.K. structures.

We show as a main result a structure theorem of compact homogeneous l.c.K. manifolds, asserting that it has a structure of a holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus. As an application of the theorem, we see that only compact homogeneous l.c.K. manifolds of complex dimension 2 are Hopf surfaces of homogeneous type. We also see that there exist no compact complex homogeneous l.c.K. manifolds; in particular neither complex Lie groups nor complex paralellizable manifolds admit their compatible l.c.K. structures.

This talk is based on a joint work with Y. Kamishima "Locally conformally Kaehler structures on homogeneous spaces" (arXiv:1101.3693).

#### Stefan Ivanov (Sofia University) Extremals for the Sobolev-Folland-Stein inequality, the quaternionic contact Yamabe problem and related geometric structures

We describe explicitly non-negative extremals for the Sobolev inequality on the quaternionic Heisenberg groups and determine the best constant in the $L^2$ Folland-Stein embedding theorem involving quaternionic contact (qc) geometry and the qc Yamabe equation. Translating the problem to the 3-Sasakian sphere, we determine the qc Yamabe invariant on the spheres. We describe explicitly all solutions to the qc Yamabe equation on the seven dimensional quaternionic Heisenberg group. The main tool is the notion of qc structure and the Biquard connection. We define a curvature-type tensor invariant called qc conformal curvature in terms of the curvature and torsion of the Biquard connection and show that a qc manifold is locally qc conformal (gauge equivalent) to the standard flat qc structure on the Heisenberg group, or equivalently, to the 3-sasakian sphere if and only if the qc conformal curvature vanishes. Possibly, this will help to reduce the qc Yamabe problem to that of the spherical qc manifolds.

#### Julien Keller (Aix Marseille University) Chow stability and the projectivisation of stable bundles

We will discuss the Chow stability of the projectivisation of a Gieseker stable bundles over a surface endowed with a constant scalar curvature Kahler metric. We will provide an example of a smooth manifold which is Chow stable but not asymptotically Chow stable. This is a joint work with J. Ross.

#### Dario Martelli (King's College, London) Geometric structures arising in string theory

Several interesting differential-geometric structures arise in the context of string theory from requiring supersymmetry. In particular, this implies the existence of spinor fields obeying certain differential equations. I will discuss the physical motivations behind these structures and I will review examples of explicit constructions, including: special holonomy manifolds, complex non-Kaehler manifolds, and Sasaki-Einstein manifolds.

#### Ruxandra Moraru (Waterloo University) Compact moduli spaces of stable bundles on Kodaira surfaces

Abstract: In this talk, I will examine the geometry of moduli spaces of stable bundles on Kodaira surfaces, which are non-Kaehler compact surfaces that can be realised as torus fibrations over elliptic curves. These moduli spaces are interesting examples of holomorphic symplectic manifolds whose geometry is similar to the geometry of Mukai's moduli spaces on K3 and abelian surfaces. In particular, for certain choices of rank and Chern classes, the moduli spaces are themselves Kodaira surfaces. This is joint work with Marian Aprodu and Matei Toma.

#### Paul Andi Nagy (Greifswald University) Symplectic forms on Kaehler surfaces

Necessary and sufficient conditions for the existence of orthogonal almost-Kaehler structures on Kaehler surfaces will be given. We will explain how these conditions work on several classes of examples. The relation to the problem of finding a symplectic form on a Kaehler surface will be outlined.

#### Stefan Nemirovski (Steklov Institute) Universal coverings of strictly pseudoconvex domains

The universal covering of a strictly pseudoconvex domain in a Stein manifold is completely determined by the local CR-geometry of its boundary. I will discuss various results and problems related to this general principle. This is joint work with Rasul Shafikov.

#### Paolo Piccinni (Sapienza Universita' di Roma) Some curiosities on Spin(9) and the sphere S^{15}

Although holonomy Spin(9) is only possible for the two 16-dimensional symmetric spaces $\mathbb OP^2$ and $\mathbb O H^2$, weakened holonomy Spin(9) conditions have been proposed and studied, in particular by Th. Friedrich. A basic problem is to have a simple algebraic formula for the canonical $8$-form $\Phi_{Spin}(9)}$, similar to the usual definition of the quaternionic 4-form $\Phi_{\mathrm{Sp}(n)\cdot \mathrm{Sp}(1)}= \omega_I^2+\omega_J^2+\omega_K^2$, witten in terms of local compatible almost hypercomplex structures (I,J,K).

In the talk, a simple formula for $\Phi_{\mathrm{Spin}(9)}$ is presented, discussing a family of local almost hypercomplex structures associated with a Spin(9)-manifold $M^{16}$. Some of these complex structures, now on model spaces $\R^{16^q}$, are then used to give an approach through Spin(9) to the very classical problem of writing down a maximal system of tangent vector fields on spheres $S^{N-1} \subset \R^N$. If time permits, some properties of manifolds equipped with a locally conformal parallel Spin(9) metric will be also discussed.

#### Sönke Rollenske (Johannes Gutenberg-Universität Mainz) Lagrangian fibrations on hyperkaehler manifolds

Hyperkaehler (also called irreducible holomorphic symplectic) manifolds form an important class of manifolds with trivial canonical bundle. One fundamental aspect of their structure theory is the question whether a given hyperkaehler manifold admits a Lagrangian fibration. I will report on a joint project with Daniel Greb and Christian Lehn investigating the following question of Beauville: if a hyperkaehler manifold contains a complex torus T as a Lagrangian submanifold, does it admit a (meromorphic) Lagrangian fibration with fibre T? I will describe a complete positive answer to Beauville's Question for non-algebraic hyperkaehler manifolds, and give explicit necessary and sufficient conditions for a positive solution in the general case using the deformation theory of the pair (X,T).

#### Yann Rollin (Nantes University) Stability of extremal metrics under complex deformations

Let $(X,\Omega)$ be a closed polarized complex manifold, $g$ be an extremal metric on $X$ that represents the K\"ahler class $\Omega$, and $G$ be a compact connected subgroup of the isometry group $Isom(X,g)$. Assume that the Futaki invariant relative to $G$ is nondegenerate at $g$. Consider a smooth family $(M\to B)$ of polarized complex deformations of $(X,\Omega)\simeq (M_0,\Theta_0)$ provided with a holomorphic action of $G$. Then for every $t\in B$ sufficiently small, there exists an $h^{1,1}(X)$-dimensional family of extremal K\"ahler metrics on $M_t$ whose K\"ahler classes are arbitrarily close to $\Theta_t$.

#### Song Sun (Imperial College London) Positivity and Sasakian geometry

We will classify compact Sasaki manifolds with positive transverse curvature. This can be viewed as a generalization of the solution to the Frankel conjecture.

#### Vladlen Timorin (HSE) Maps that take lines to conics

We will discuss generalizations of the classical theorem of Moebius (1827): a one-to-one self-map of a real projective space that takes all lines to lines is a projective transformation. E.g. we study sufficiently smooth local maps taking line segments to parts of conics. A description of local maps taking line segments to circle arcs depends non-trivially on the dimension (the descriptioninvolves classical geometries, quaternionic Hopf fibrations, representations of Clifford algebras). For most dimensions, it is still missing.

#### Valentino Tossati (Columbia University) Collapsing of abelian fibred Calabi-Yaus and hyperkahler mirror symmetry

We will address the problem of understanding the collapsing of Ricci-flat Kahler metric on abelian fibred projective Calabi-Yau manifolds. We will then explain an application of these results to the Strominger-Yau-Zaslow picture of mirror symmetry for some hyperkahler manifolds. Joint work with Mark Gross and Yuguang Zhang.

#### Victor Vuletescu (University of Bucharest) Birational aspects in the study of LCK manifolds

We study the behaviour of locally conformally Kahler (LCK for short) manifolds under birational transformations. We show that the blow-up of an LCK manifold X along a subvariety Y is LCK iff Y is globally conformallly Kahler (GCK). Using the same methods, we also show that a twistor space is LCK iff it is GCK. We will also adress a number of open questions.

This is joint work with L. Ornea and M. Verbitsky.

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